L(s) = 1 | + (−1.38 − 0.270i)2-s + (0.869 − 2.09i)3-s + (1.85 + 0.751i)4-s + (1.95 − 3.84i)5-s + (−1.77 + 2.67i)6-s + (0.861 + 3.58i)7-s + (−2.36 − 1.54i)8-s + (−1.52 − 1.52i)9-s + (−3.75 + 4.80i)10-s + (−1.33 + 1.56i)11-s + (3.18 − 3.23i)12-s + (−0.539 − 0.880i)13-s + (−0.225 − 5.21i)14-s + (−6.35 − 7.44i)15-s + (2.87 + 2.78i)16-s + (0.209 + 2.66i)17-s + ⋯ |
L(s) = 1 | + (−0.981 − 0.191i)2-s + (0.501 − 1.21i)3-s + (0.926 + 0.375i)4-s + (0.875 − 1.71i)5-s + (−0.724 + 1.09i)6-s + (0.325 + 1.35i)7-s + (−0.837 − 0.545i)8-s + (−0.509 − 0.509i)9-s + (−1.18 + 1.51i)10-s + (−0.403 + 0.472i)11-s + (0.920 − 0.934i)12-s + (−0.149 − 0.244i)13-s + (−0.0601 − 1.39i)14-s + (−1.64 − 1.92i)15-s + (0.717 + 0.696i)16-s + (0.0509 + 0.647i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0267 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0267 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.695691 - 0.714587i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.695691 - 0.714587i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.38 + 0.270i)T \) |
| 41 | \( 1 + (-6.09 - 1.95i)T \) |
good | 3 | \( 1 + (-0.869 + 2.09i)T + (-2.12 - 2.12i)T^{2} \) |
| 5 | \( 1 + (-1.95 + 3.84i)T + (-2.93 - 4.04i)T^{2} \) |
| 7 | \( 1 + (-0.861 - 3.58i)T + (-6.23 + 3.17i)T^{2} \) |
| 11 | \( 1 + (1.33 - 1.56i)T + (-1.72 - 10.8i)T^{2} \) |
| 13 | \( 1 + (0.539 + 0.880i)T + (-5.90 + 11.5i)T^{2} \) |
| 17 | \( 1 + (-0.209 - 2.66i)T + (-16.7 + 2.65i)T^{2} \) |
| 19 | \( 1 + (-0.0153 - 0.00939i)T + (8.62 + 16.9i)T^{2} \) |
| 23 | \( 1 + (3.82 + 2.78i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (0.347 - 4.41i)T + (-28.6 - 4.53i)T^{2} \) |
| 31 | \( 1 + (2.93 - 9.03i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-1.09 - 3.36i)T + (-29.9 + 21.7i)T^{2} \) |
| 43 | \( 1 + (0.739 + 4.66i)T + (-40.8 + 13.2i)T^{2} \) |
| 47 | \( 1 + (-0.161 + 0.671i)T + (-41.8 - 21.3i)T^{2} \) |
| 53 | \( 1 + (-0.621 - 0.0489i)T + (52.3 + 8.29i)T^{2} \) |
| 59 | \( 1 + (1.67 - 2.31i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (0.0314 - 0.198i)T + (-58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (-7.84 + 6.69i)T + (10.4 - 66.1i)T^{2} \) |
| 71 | \( 1 + (-1.54 - 1.31i)T + (11.1 + 70.1i)T^{2} \) |
| 73 | \( 1 + (-3.34 + 3.34i)T - 73iT^{2} \) |
| 79 | \( 1 + (1.95 + 0.808i)T + (55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + 14.8iT - 83T^{2} \) |
| 89 | \( 1 + (-3.06 + 0.735i)T + (79.2 - 40.4i)T^{2} \) |
| 97 | \( 1 + (2.16 - 1.84i)T + (15.1 - 95.8i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.45254832986822154342222344745, −12.11321771295241024790733541462, −10.30419249337023911872089349476, −9.108517930607833784644364246890, −8.572108600871949728680335988246, −7.84414718189301181945554711806, −6.32205994997838355357454703963, −5.20322546242777927968230335057, −2.34254142311087673456067218046, −1.51071304293979501957969435105,
2.51814976894510888806584606415, 3.82452813826489011027915254907, 5.80720161772746564609044811419, 7.04049635538343498614519101521, 7.87239018544449087378690349539, 9.658263669442254186259489901891, 9.796657810909821967163262959457, 10.87553337134434097809361427859, 11.19279721238121843080644627945, 13.66350458507695436375206683746